IN ad-hoc wireless networks, network nodes are independent,

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 Altruistic Coalition Formation in Cooperative Wireless Networks Mohammed W. Baidas, Member, IEEE and Allen B. MacKenzie, Senior Member, IEEE Abstract In this paper, altruistic coalition formation in cooperative relay networks is studied. The communication is performed over two phases, the broadcasting phase and the cooperation phase. In the broadcasting phase, each node broadcasts its signal in its time-slot, while in the cooperation phase, all the nodes within their coalitions simultaneously relay each others signals. A distributed merge-and-split algorithm is proposed to allow nodes to form coalitions and improve their total achievable rate. Moreover, the impact of different power allocation criteria is studied, where the sum-of-rates maximizing power allocation is shown to promote altruistic coalition formation and results in the largest coalitions among the different power allocation criteria. Finally, the proposed algorithm is compared with centralized power allocation and coalition formation, and shown to yield a good tradeoff between network sum-rate and computational complexity. Index Terms Coalition formation, cooperation, decode-andforward (DF), network coding, power allocation. I. INTRODUCTION IN ad-hoc wireless networks, network nodes are independent, autonomous and selfish by nature and thus may not voluntarily share their transmission resources with other nodes. In other words, there is an element of competition and selfishness since all participating network nodes desire to maximize their utilities by maximizing their share of transmission resources. Moreover, randomly distributed nodes with local information may not know whom to cooperate with even if they are willing to cooperate. Although cooperative communications have been shown to yield significant performance gains [1], cooperation entails several costs, such as bandwidth and power. Ignoring such costs is unwarranted as it may severely affect the nodes own performance. Particularly, network nodes may not cooperate and instead divert their resources to direct data transmissions. Alternatively, a group of nodes could form a coalition and cooperate to maximize the overall gains of the group and thus promote altruism. Specifically, each node seeks partners to form a cooperative coalition to achieve rate improvement for itself and/or for the whole coalition. In such a case, the greatest immediate benefits may not be achieved by the nodes that bear the greatest costs. Establishing cooperation in wireless ad-hoc networks without a centralized controller is a dynamic process. Hence, designing The associate editor coordinating the review of this paper and approving it for publication was Prof. H. Li. Manuscript received September 11, 212; revised February 5, May 25, July 27 and September 25, 213. M. W. Baidas is with the Electrical Engineering Department, Kuwait University, Kuwait ( baidas@ieee.org). A. B. MacKenzie is with Virginia Tech, Bradley Department of Electrical and Computer Engineering, Virginia Tech, USA ( mackenab@vt.edu). Digital Object Identifier 1.119/TCOMM /1$25. c 213 IEEE practical distributed algorithms that can promote cooperation without relying on centralized control is a highly desirable but considerably difficult task. Coalitional game theory has emerged as an effective mathematical tool for modeling users cooperation and designing distributed protocols in wireless networks. Several works have considered coalitional formation for user cooperation in wireless networks. For instance, a simple and distributed merge-and-split algorithm is proposed in [2] for the formation of virtual multiple input multiple output (MIMO) clusters of selfish single-antenna nodes. In [3], the stability of the grand coalition of transmitter and receiver cooperation in an interference channel is studied for both flexible transferable and non-transferable apportioning schemes. The curse of the boundary nodes in selfish packet-forwarding wireless networks is resolved using coalitional games in [4]. In [5], fair group coalitions for power-aware routing in wireless networks is studied and distributed algorithms based on max-min fairness are proposed. Distributed coalitional formations with transferable utilities and stable outcomes in relay networks are studied in [6]. In [7], a coalition formation game for relay transmission is studied based on a Markov-chain model, where the nodes are assumed to be selfish and aim at maximizing their individual throughput. Furthermore, stability analysis of the coalition formation process under system parameters uncertainty is also investigated. A distributed coalition formation strategy for collaborative sensing tasks in camera sensor networks has been studied in [8,9]. Specifically, the authors propose a model that supports task-driven node selection and aggregation that is based on local decision-making and internode communication, and provides robustness and scalability. In this work, altruistic coalition formation is particularly considered and the aim is to address the following questions: (1) How can coalitions be formed in a distributed fashion?, (2) What is the impact of different power allocation criteria on coalition formation?, and (3) What is the effect of mobility on the coalition formation process? To form cooperative groups, a coalition formation algorithm based on merge-andsplit rules is proposed and proven to converge with arbitrary merge-and-split iterations. Each network node is treated as a player, who seeks partners to form a cooperative group to improve its transmission rate and/or that of the whole group through spatial diversity while incurring some power cost to meet a target SNR for information exchange. This in turn suggests a tradeoff between the gains and costs of cooperation. Furthermore, as the size of the cooperative group increases, both the gain and the cost also increase to the point where adding an additional node results in a cost that outweighs

2 2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION the diversity gain. Additionally, since network nodes can be either mobile or static, the aim here is to analyze the long-term behavior of network coalition formation when the nodes are mobile within the network area as opposed to their coalition formation when they are static (i.e. at fixed locations). To the best of the authors knowledge, no existing work has employed coalitional games in the analysis and design of algorithms for altruistic coalition formation in network-coded cooperative wireless networks 1. The main contributions of this paper are summarized as follows: Modeled network-coded cooperative transmissions as a coalitional game in partition form with non-transferable utility and preference relation based on utilitarian order. Proposed a distributed merge-and-split algorithm for altruistic coalition formation in ad-hoc wireless networks. Additionally, the convergence, partition stability and complexity properties of the proposed distributed algorithm have been studied. Evaluated the impact of different power allocation criteria on coalition formation, where the sum-of-rates maximizing power allocation has been shown to promote altruistic coalition formation and result in the largest coalitions among the different power allocation criteria. Evaluated the proposed algorithm and demonstrated that it can achieve a network sum-rate that is comparable with that of a centralized algorithm; but with less computational complexity. Additionally, it has been shown to efficiently adapt to nodes mobility and network topology changes. In the remainder of this paper, the system model is presented in Section II. In Section III, the coalitional formation framework is discussed, while the proposed distributed coalition formation algorithm is provided in Section IV. The different cooperative power allocation criteria are discussed in Section V, while the convergence, stability and complexity properties of the proposed distributed algorithm are discussed in Section VI. The numerical results are presented in Section VII, while Section VIII discusses some related practical issues. Finally, conclusions are drawn in Section IX. II. SYSTEM MODEL Consider an ad-hoc wireless network that consists of N single-antenna half-duplex decode-and-forward nodes which are denoted S 1,S 2,...,S N for N 3. Each node wishes to exchange its data symbol x j for j {1, 2,...,N} with a common destination node D. The channel between nodes S j and S i is given by h j,i = e jθj,i d ν j,i, with ν being the path-loss exponent and θ j,i is the signal s phase uniformly distributed in the interval [, 2π] while d j,i is the distance between the two nodes [2]. Also, the channel h j,i between nodes S j and S i is assumed to be reciprocal (i.e. h j,i = h i,j ) with perfect channel estimation at each node. 1 The material in this paper has been presented in part at the 8th IEEE International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Oct. 212 [1]. In this correspondence, the system model is extended, convergence and stability properties of the proposed coalition formation algorithm are thoroughly studied and the effect of mobility on coalition formation is considered. Fig. 1. Example of Cooperative Coalitions and Their Transmissions Further, let S = {S 1,S 2,...,S N } be the finite, non-empty set of all network nodes that eventually self-organize into K (for 1 K N) mutually exclusive coalitions of cooperative nodes C = {C 1,C 2,...,C K } with no cooperation between coalitions. Also, let C k S denote a coalition with C k nodes (where represents the cardinality of a set) and 1 C k N. An individual non-cooperative player is called a singleton coalition while the set S is called the grand coalition when all the N network nodes cooperate. An example of a network of N =5nodes with a possible coalition formation is illustrated in Fig. 1. The communication between each node and the destination is performed in a TDMA fashion over N +1 time-slots and is split into two phases, namely the broadcasting phase and the cooperation phase. A. Broadcasting Phase In the broadcasting phase of N time-slots, each node S j in its time-slot T j broadcasts its data symbol x j, which is received by the N 1 other nodes S i in the network for i {1, 2,...,N} i j, and the destination. The signal received at node S i for i j is expressed as y j,i = P Bj h j,ix j + n j,i, (1) while the received signal at the destination is given by y j,d = P Bj h j,d x j + n j,d, (2) where P Bj is the transmit power in the broadcasting phase at node S j, and n j,i and n j,d are zero-mean complex additive white Gaussian noise (AWGN) samples with variance N,at node S i and the destination, respectively. Upon the completion of the broadcasting phase, each node S i will have received a set of N 1 signals {y j,i} N j=1,j i comprising symbols x 1,...,x i 1,x i+1,...,x N from all the other nodes in the network. In addition, the destination will have received N signals {y j,d } N j=1. Each node S i then performs a matched filtering operation on each of the received signals y j,i and the signal-to-noise ratio (SNR) at the output of the matched-filter is given by [1] γ BP j,i = PB j hj,i 2 N = PB j d ν j,i N. (3) Let f j denote the node farthest from S j in S j s coalition C k. Each node S j C k broadcasts its symbol using transmit power P Bj required to maintain a target SNR of γ between itself and node S fj as

3 BAIDAS and MACKENZIE: ALTRUISTIC COALITION FORMATION IN COOPERATIVE WIRELESS NETWORKS 3 P Bj γn d ν j,f j, (4) where it is assumed that γ is common to all the nodes in the network. Clearly, there is a tradeoff between the power invested in satisfying the target SNR and power allocated to the other members in coalition C k. It should be noted that each node has a transmit power constraint of P that is shared between the two transmission phases as given by P = P Bj + P Cj, where P Cj is the effective cooperative power at node S j to relay the symbols of the other nodes in coalition C k. Specifically, P Cj =max [, min(p P Bj,P) ], with P Cj = S i C k,i j P C i,j, and P Ci,j is the cooperative power node S j utilizes in relaying node S i s symbol x i to the destination, for i j. B. Cooperation Phase In the cooperation phase, each node S i for S i C k, C k C and C k 2 in time-slot T N+1 forms a linearly-coded signal X i (t) of the C k 1 received signals from the nodes in C k, during the broadcasting phase. For multiuser detection at the destination, each decoded symbol x l at node S i is spread using a signature waveform c l (t), where it is assumed that each node S i (for i l) and the destination know the signature waveforms of all the other nodes in the coalition. The cross-correlation of c l (t) and c i (t) is ρ l,i = c l (t),c i(t) (1/T T s s) c l (t)c i (t)dt for l i with ρ i,i =1and T s being the symbol duration. Thus, the transmitted signal by node S i is X i(t) = P Cl,i x l c l (t). (5) S l C k,l i The received signal at the destination assuming perfect timing synchronization is written as N Y d (t) = h i,d X i(t)+n d (t), (6) i=1 where n d (t) is the AWGN process at the destination. By substituting (5) into (6), Y d (t) is re-written as Y d (t) = K k=1 S l C k S i C k,i l P Cl,i h i,d x l c l (t)+n d (t). (7) At the destination, multiuser detection is performed on Y d (t) to extract symbol x j of node S j C k, C k C. Specifically, Y d (t) is passed through a matched filter bank (MFB), yielding Y j,d = Y d (t),c j(t) = K k=1 S l C k α l x l ρ l,j + n j,d, (8) where n j,d is a zero-mean AWGN noise sample with variance N, while α l is given by α l = P Cl,i h i,d. (9) S i C k,i l The output of the MFB is expressed in vector form of N signals as Y d = RAx + n n n d, where Y d =[Y 1,d,...,Y N,d ] T, x = [x 1,...,x N ] T, and n n n d =[ n 1,d,..., n N,d ] T CN(,N R). Also, R and A are N N matrices, where R is given by 1 ρ 1,2 ρ 1,N ρ 2,1 1 ρ 2,N R = , (1) ρ N,1 ρ N,2 1 while the diagonal matrix A is expressed as A = diag [α 1,α 2,...,α N ]. The received signal vector Y d is then decorrelated (assuming matrix R is nonsingular) as ỸỸỸ d = R 1 Y d = Ax +ñññ d, where ñññ d = R 1 n n n d and ñññ d CN (,N R 1). It is assumed that ρ l,j = ρ, l j. Therefore, the decorrelated received signal is obtained as Ỹ j,d = α jx j +ñ j,d, (11) where ñ j,d CN(,N ϱ N ), and ϱ N is given by 1+(N 2) ρ ϱ N = 1+(N 2) ρ (N 1) ρ. (12) 2 The received instantaneous SNR of node S j s symbol (where S j C k ) at the destination is given by γ j = γj BP +γj CP, where γj BP is expressed in (3), and γj CP is obtained by passing Ỹj,d through a matched-filter. Therefore, γ j is obtained as [11] γ j = PB h j j,d 2 + P Cj,i h i,d 2. (13) N N ϱ N S i C k,i j Upon the completion of the broadcasting and cooperation phases, the destination will have received C k independent copies of symbol x j of node S j C k and thus achieving a diversity order of C k [1]. III. COALITION FORMATION FRAMEWORK Let υ j (C k ) denote the payoff of each node S j in coalition C k, which is assumed to be equal to the achievable rate. Based on the discussed system model, a singleton coalition of node S j occurs when it does not form a cooperative coalition with other nodes. In this case, node S j utilizes all its available power P and transmits its data once every N +1 time-slots. Thus, the payoff of node S j is obtained as υ j({s j}) =Rj,d D = 1 N +1 log 2 (1+ P h ) j,d 2, (14) where Rj,d D is the achievable rate with direct transmission. Additionally, Rj,d D represents the non-cooperative payoff of any node S j for S j S. On the other hand, for coalition C k with C k 2, the achievable rate of node S j due to the cooperative transmission is given by Rj,d C = 1 N +1 log 2 1+ PB h j j,d 2 + P Cj,i h i,d 2. N N S i C k,i j ϱ N (15) Therefore, the payoff of each node S j in coalition C k is given by υ j (C k )=Rj,d C, S j C k. Consequently, the value of a coalition is υ(c k )= υ j(c k ), (16) S j C k N which is equivalent to the sum-rate of the coalition. Definition 1: A coalition game is said to have a nontransferable utility (NTU) if the coalition value cannot be arbitrarily apportioned among its nodes and each node will have its own value within a coalition.

4 4 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION V(C k )= { } υ(c k ) R Ck S j C k,υ j(c k )=Rj,d C, if P Cj >, and υ j(c k )= Rj,d, D otherwise.. (17) Based on the proposed system model, the coalition game in hand has a non-transferable utility, as a specific achievable rate for each node in a coalition is achieved [12]. In addition, the coalition game formulated in this work is in the characteristic function form. That is, utilities achieved by the players in a coalition are unaffected by those outside it. Definition 2: A coalitional game with non-transferable utility is defined by a pair (S, V), where S is a finite set of N players, and V is a set valued function such that for every coalition C k S, V(C k ) is a closed convex subset of R C k that contains the payoff vectors the players in C k can achieve. In the proposed system model, V : C k R Ck such that V(φ) = φ, and if C k φ, then V(C k ) is non-empty and closed. Moreover, the coalitional set-valued function V of a coalition C k Sis defined as in (17). Note that P Cj = if and only if P Bj = P, which implies that node S j has no interest in cooperation. Remark 1: The proposed network-coded transmission model is a coalitional game (S, V) in partition form with nontransferable utility, where V(C k ) is a singleton set, as defined by (17), and is thus a closed and convex subset of R Ck. Remark 2: In the proposed NTU coalitional game (S, V), the grand coalition of all the nodes rarely forms due to the target SNR power costs. Based on the SNR target defined in (4), the power cost for coalition formation depends on the distance between the network nodes which in turn governs coalitions sizes. As the size of a coalition increases, the cooperative gain and power cost per node also increase (i.e. cooperation gains in a coalition are limited by power costs). However, the power saving due to the diversity gains gradually diminish even with the increase in the number of cooperative nodes in a coalition, at which point, no additional nodes should join the coalition. This prevents cooperative networks from forming a grand coalition and instead form independent disjoint coalitions. In turn, the proposed game is modeled as a coalition formation game, with the aim of finding the network s coalitional structure [12]. It is noteworthy that solution concepts for coalitional games based on the core are not applicable to the game model in hand due to the power costs. Additionally, it should be noted that power costs at each node forming a coalition are based on the following. First, to cooperate, the node must allocate power to the cooperation phase, which limits the transmit power available for broadcasting of node s own symbols. Second, cooperation with the nodes in a given coalition imposes a lower-bound on the broadcasting power (to ensure target SNR is met within the coalition). IV. DESIGN OF DISTRIBUTED COALITION FORMATION ALGORITHM In this section, the aim is to study how coalitions can be formed in a distributed manner. Specifically, a coalition is formed if it is beneficial to at least one node in the coalition and also for the coalition as a whole. Also, the nodes of a coalition can avoid merging with other coalitions if they are as well off as a result of not merging. Furthermore, when nodes form a coalition, they cannot unilaterally deviate on their own. In turn, coalition structure changes are determined by the members of a coalition interacting with one another as a unit. Since network nodes are rational and autonomous, the design of an iterative distributed algorithm to form a network coalition structure that improves that network sum-rate is highly desirable. But first, several concepts must be defined. Definition 3: A collection of coalitions, denoted as C, is defined as C = {C 1,C 2,...,C K } for 2 K N mutually disjoint coalitions C k of C. Equivalently, a collection is any arbitrary group of disjoint coalitions C k of C that does not necessarily span all the players of S. If a collection spans all the players in S (i.e. K k=1 C k = S), the collection is a partition of S. Definition 4: A preference operator is defined for comparing two collections Q = {Q 1,...,Q l }, and R = {R 1,...,R p } that are partitions of the same subset A S (i.e. same players in Q and R). Thus, Q R implies that the way Q partitions A is preferred to the way R partitions A. In the coalitional game theory literature, comparison relations based on orders are split into two categories [13]: individual value orders and coalition value orders. In the former category, comparison is performed on the basis of individual payoffs (e.g. Pareto order). Specifically, no node is willing to form a coalition with other nodes if forming that coalition would result in a transmission rate that is less than its direct transmission rate (i.e. selfish behavior). In the latter category, two collections (or partitions) are compared based on the value of the coalitions inside these collections, such as the utilitarian order (e.g. Q R = l i=1 υ(q i) > p i=1 υ(r i)). Particularly, a node will form a coalition with other nodes even if its achievable cooperative transmission rate is less than its direct transmission rate, as long as the sum-rate of the coalition is improved and is greater than the sum of the direct transmission rates of all the nodes within the coalition (i.e. altruistic behavior). In this work, the utilitarian order comparison relation is assumed as it is more suited to the studied altruistic coalition formation. There are two successive rules for forming and breaking coalitions, known as merge-and-split rules [13]. Definition 5 (Merge Rule): Merge any collection of disjoint coalitions {Q 1,...,Q l }, where { l k=1 Q k} {Q 1,...,Q l }, thus {Q 1,...,Q l } { l k=1 Q k}. Definition 6 (Split Rule): Split any coalition { l k=1 Q k}, where {Q 1,...,Q l } { l k=1 Q k}, thus { l k=1 Q k} {Q 1,...,Q l }. The merge-and-split rules simply mean that two (or more) coalitions will merge if their merger would do more good than harm to the overall coalition value (or equivalently, sum-rate) of the merged coalition. Otherwise, coalitions will split into smaller ones or even singletons. As noted earlier, most cooperative communication systems proposed in the literature are based on the assumption of fully cooperative behaviors; while ignoring cooperation costs. Such costs not only limit the benefits of cooperation but also impair the user s performance. Additionally, most of the

5 BAIDAS and MACKENZIE: ALTRUISTIC COALITION FORMATION IN COOPERATIVE WIRELESS NETWORKS 5 current research consider selfish nodes; however, in this work, the case of altruistic nodes that aim at improving the network sum-rate is considered. Therefore, there is a need for deriving a practical distributed algorithm that promotes cooperation, takes into account the tradeoff between cooperation gains and costs, and eliminates the need for centralized entities (as in the case of ad-hoc wireless networks). Due to the fact that network nodes are geographically distributed, then nodes must be able to self-organize into a stable network partition and adapt this structure to topology changes and mobility, even in the absence of a centralized controller. For our coalitional game-theoretic algorithmic design, the utility of each node has been defined as its achievable rate, and cooperation costs have been taken into account in the form of the transmit power need to satisfy a target SNR each neighboring potential node. Additionally, the novel concept of merge-and-split will be utilized to allow network nodes to form altruistic coalitions based on the utilitarian order. Specifically, the merge-and-split processes will be based on the sum-rate improvement achievable through altruistic coalition formation. A. Algorithm Description The network operation starts at τ =with network nodes being partitioned into singleton coalitions (i.e. C j = {S j } for 1 j N such that C = {{S 1 }, {S 2 },...,{S N }}) and each node S j determines its achievable direct transmission rate Rj,d D. After that, the following three phases take place. 1) Node Discovery: Each node S j S discovers the neighboring potential nodes with which it can possibly merge using P Bj = P. Specifically, for each node S j, potential partners lie within a circle with radius determined by the power P γn d ν j,f j required for symbols exchange while meeting the target SNR γ. Thus, if the received signal at node S i satisfies γ, it is considered to be decoded correctly. Let D j be the set of nodes that decoded node S j s symbol correctly, i.e. D j = { S i S and i j : γj,i BP γ}. (18) After that, node S j broadcasts a request-to-send (RTS) message which is received by all the nodes in D j. Then, each node S i D j replies to node S j with a clear-to-send (CTS) message that contains its channel state information with the destination. If the decoding set of node S j is empty (i.e. D j = φ), then it employs direct transmission and does not form a coalition with any other node. Otherwise, node S j enumerates all the possible distinct coalitions of S j D j. In the case of a coalition C k, the potential nodes lie within the intersection of C k circles, each centered around node S j C k. Clearly, the node discovery phase significantly reduces the coalition formation space. 2) Adaptive Coalition Formation: In this phase, the timeindex is updated to τ = τ +1 and each node sequentially proposes to merge with one of its potential partners. If such a merge is desirable by all the nodes according to the utilitarian order, then a coalition with one or more of the potential nodes could form by a merge agreement of all the participating nodes. For all merged coalitions, a random node is elected as a coalition-head [14], which is responsible for periodically exchanging timing information with the rest of the coalition 2. After that, the power allocation fractions of each node are determined according to one of the power allocation criteria discussed in Section V. After all the nodes have made their merge decisions, the merge process ends, resulting in a partition M τ = Merge(C τ 1 ). If the sum-rate value a group of nodes achieved by forming a coalition is less than the value achieved before the merger, they split into singletons or coalitions of smaller sizes. At the end of the split process, a partition C τ = Split(M τ ) is obtained. A sequence of merge-and-split processes along with time-index updates take place in a distributed manner via appropriate control channels, depending on the achievable rate improvement of each node and coalition, until there is no need for any merging/splitting in the current partition, in which case the final partition C = C τ is obtained. It should be noted that in the node discovery phase, enumerating all distinct coalitions does not necessarily require significant computations. Even if the number of possible distinct coalitions is large, each node in the adaptive coalition formation phase takes turn in submitting merge proposals to other potential nodes. So, it is likely that other potential nodes may submit merge proposals. In other words, each node will not necessarily propose each possible coalition to its potential nodes as some of these nodes will propose them. Additionally, a node may not necessarily have to enumerate all possible distinct coalitions; on the contrary, a node might start enumerating only the small-sized potential coalitions and not have to propose larger coalitions if a sum-rate improvement is achieved with small coalitions. In other words, a complete enumeration of potential coalitions should not be required as the process can stop as soon as a candidate merger is identified. 3) Data Transmission: In this final phase, data transmission of each node takes place in the form of broadcasting and cooperation, over a total of N +1 time-slots and as described in Section II. Finally, the above three phases repeat in response to topology changes or mobility, as discussed later. The network initialization and proposed distributed merge-andsplit coalition formation algorithm are summarized in Table I. It should be noted that the resulting partition from the proposed merge-and-split algorithm is not guaranteed to be optimal (i.e. is not the one that maximizes the network sumrate). This is because the formed coalitions do not exchange information about their values and thus have no way of knowing whether there are different partitions that could lead to better network sum-rate. Even if all coalition values are known, no known algorithm can determine the optimal partition with time complexity that is polynomial in the number of possible coalitions [17]. V. IMPACT OF DIFFERENT POWER ALLOCATION CRITERIA It is intuitive to note that network coalition formation is dependent on the cooperative power allocation within each coalition. Therefore, the following power allocation criteria are studied. 2 The nodes transmission within a coalition can be assumed to be perfectly synchronized during the cooperation phase [15] [16].

6 6 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION TABLE I NETWORK INITIALIZATION AND PROPOSED DISTRIBUTED MERGE-AND-SPLIT COALITION FORMATION ALGORITHM Initial State: At the beginning of all time, initialize time-index at τ = with the network being partitioned as C ={{S 1}, {S 2},...,{S N }}. Coalition Formation Algorithm: Phase 1 - Node Discovery: Each node determines its neighbors and potential coalitions. Phase 2 - Adaptive Coalition Formation: Coalition formation using merge-and-split rules occurs. repeat (a) Update time-index: τ = τ +1. (b) M τ = Merge(C τ 1 ): coalitions in C τ 1 make merge decisions based on the merge rule. (c) C τ = Split(M τ ): coalitions in M τ make split decisions based on the split rule. until merge-and-split terminates with final partition denoted C. Phase 3 - Data Transmission: Each node transmits its data symbol in the broadcasting phase and the nodes within every coalition relay data for each other during the cooperation phase. A. Equal Power Allocation (EPA) Under this criterion, a node S i C k determines its maximum required broadcasting power as P Bi = max{γn / h j,i 2 } Sj C k,j i and then the cooperative power P Ci = P P Bi is equally allocated to the other nodes in C k in the form of (EPA) : P Cj,i = P PB i C k 1, Sj C k, and j i. (19) In this case, each node naively allocates its remaining power equally to the other nodes in C k. B. Sum-of-Rates Maximizing Power Allocation (SRM-PA) The sum-of-rates maximizing power allocation problem of coalition C k is solved by the coalition-head and is expressed as (SRM-PA): max S i C k Ri,d C s.t. P Bi + P Cj,i P, S i C k, (2a) S j C k,j i P Bi γn / h j,i 2, S j,s i C k and j i, (2b) P Bi, S i C k, (2c) P Cj,i, S j,s i C k and j i. (2d) The first constraint in (2) enforces the total power constraint, while the second constraint ensures that the target SNR is met by each node S i C k. The last two constraints impose the non-negativity of the allocated broadcasting and cooperative powers, respectively. C. Max-Min Rate Power Allocation (MMR-PA) The power allocation problem under the max-min rate fairness criterion for coalition C k solved by the coalition-head is expressed as (MMR-PA): max η s.t. η Ri,d C, S i C k, (21a) P Bi + S j C k,j i P Cj,i P, S i C k, (21b) P Bi γn / h j,i 2, S j,s i C k and j i, (21c) P Bi, S i C k, (21d) P Cj,i, S j,s i C k and j i. (21e) The first constraint imposes max-min rates while the rest of the constraints are as in problem (2). Problems SRM-PA and MMR-PA can be verified to be convex [18] and thus can be solved efficiently by any standard convex optimization algorithm [19]. Therefore, solving such problems at a coalitionhead should pose no severe computational overhead. Remark 3: Since the achievable rate of each node in a coalition C k is strictly monotonically increasing in the allocated power, the total power constraint is always met (i.e. P Bi + S j C k,j i PC = P, j,i Si C k and C k C). VI. CONVERGENCE, PARTITION STABILITY AND COMPLEXITY In this section, the convergence, partition stability, and complexity properties of the proposed algorithm are studied. A. Convergence We first establish the convergence of the coalition formation process. Note that similar results are proved in [13] and [2]. Theorem 1 (Convergence): The coalition formation process based on the merge-and-split rules converges in a finite number of iterations. Proof: The merge-and-split rules under the utilitarian order only permit mergers and splits if they strictly improve the total value of the coalitions. Thus, the total value of the coalition partition is strictly increasing in the coalition formation process. Since the number of possible partitions is finite, the total value is bounded and the process must converge in a finite number of iterations. B. Partition Stability Stable coalition structures in coalition formation games correspond to an equilibrium state in which users do not have incentives to leave the already formed coalitions. There are a number of possible notions of stability, and we follow [13] (and [2], which uses slightly different terminology) in characterizing these notions with respect to a defection function. Definition 7 (D(P) Function): A defection function D is a function which associates with each partition P of S a group of collections in S [13]. In words, D(P) consists of all collections C = l {C 1,C 2,...,C l } such that the players in C, j=1 C j, can leave their current coalitions (specified by the partition P) to form the new coalitions specified by C. Definition 8 (C in the frame of P): Given a collection C = {C 1,C 2,...,C l } and a partition P = {P 1,P 2,...,P k } of S, we define C in the frame of P as { } l l l C[P] = P 1 C j,p 2 C j,...,p k C j \. j=1 j=1 j=1 (22)

7 BAIDAS and MACKENZIE: ALTRUISTIC COALITION FORMATION IN COOPERATIVE WIRELESS NETWORKS 7 That is, C[P ] is a collection consisting of the same elements as in the collection C, divided according to the partition P [13]. Definition 9 (D-Stability): A partition P = {P 1,P 2,...,P k } of S is D-stable if C[P] C for all C D(P) such that C[P] C [13]. In particular, we describe two important defection functions and their corresponding stability notions [13]. The D c defection function associates with each partition P of S the group of all collections in S. As such, this function allows any group of players to leave the partition P through any operation and create an arbitrary collection in S. As a result of this complete flexibility, the resulting D c -stability is the strongest notion of stability. Definition 1 (D c (P) Function): For each partition P, D c (P) is the family of all collections of S [13]. The second defection function, D hp, associates with each partition P of S the group of all partitions of S that can form by merging or splitting the coalitions in P. Definition 11 (P-Compatibility): A coalition T of S is called P-compatible if for some i {1,...,K}, T P i, and P-incompatible, otherwise [13]. Definition 12 (P-Homogeneity): A partition Q = {Q 1,Q 2,...,Q l } of S is called P-homogeneous if for each j {1,...,l} there exists some i {1,...,k} such that either Q j P i or P i Q j. So, any P-homogeneous partition arises from P by allowing each coalition either to split into smaller coalitions or to merge with other coalitions (or to remain unchanged) [2]. Definition 13 (D hp (P) Function): A function that associates with each partition P of S the family of all P- homogeneous partitions of S [2]. Theorem 2 (D hp -Stability): A partition is D hp -stable if and only if it is the outcome of iterating the merge-and-split rules [2]. Proposition 1: The final partition C that results from our proposed algorithm is D hp -stable. Proof: This is an immediate result of Theorem 2. Consequently, to find a D hp -stable partition, it suffices to iterate the merge-and-split rules starting from any initial network partition until partition C is reached, in which case players have no incentive to leave partition C through mergeand-split to form other partition in S. Theorem 3 (D c -Stability): If P is a D c -stable partition then P is unique, and P is the outcome of every iteration of the merge and split rules [13]. Proposition 2: The proposed coalition formation algorithm converges to the unique D c -stable partition, if such a partition exists. Otherwise, the proposed merge-and-split algorithm results in a D hp -stable network partition. Proof: This is an immediate consequence of Theorem 3 and Proposition 1. Remark 4: If a D c -stable partition exists, then it will be the unique network sum-rate maximizing partition, and it will be the outcome of the merge-and-split algorithm starting from any initial network partition. It is noteworthy that our simulation results suggest that usually no D c -stable partition exists (as will be further discussed in the penultimate paragraph of Section VII). C. Complexity Analysis 1) Communication Complexity: The communication complexity of the proposed algorithm is related to the number of merge-and-split operations, which is directly related to the total number of coalition formation proposals M sent by each of the N nodes. Two extreme cases are considered: (1) if all the proposals are rejected, and (2) if all the proposals are accepted. In the first case and as described in Section IV-A, each node S i S submits at most D i proposals, where D i N 1. Now, if the first node submits N 1 proposals and the second submits N 2 proposals and so on, then the total number of proposals is M worst = N 1 i=1 i = 1 2 N(N 1) [2]. Thus, in the worst case, the complexity is of the order O(N 2 ). In the second case where all the proposals are accepted, the total number of proposals is only M best = N, and a complexity order of O(N). In practice, the number of proposals is between these two extreme cases (i.e. M best M M worst ). In fact, the number of proposals is much lower than 1 2N(N 1) as the proposed algorithm tends to merge the smaller coalitions first and then the bigger ones but with reduced possibilities. Hence, if L messages are required per coalition formation proposal, then L M messages are required until convergence of the algorithm. 2) Computational Complexity: An equally important factor into the operation of the distributed merge-and-split coalition formation algorithm is the computational complexity involved in the cooperative power allocation. As for the equal power allocation criterion, the calculation of power allocation at each node is trivial (i.e. with negligible computational complexity). However, for the sum-of-rates maximizing and max-min rate power allocation criteria, the computational complexity is dependent on the number of nodes in each coalition (which determines the number of variables and constraints). Despite the fact that such problems are convex and thus can be computed efficiently, doing so repetitively may impose significant overhead and delay to each coalition-head, especially for potentially large coalition sizes. In other words, the computational complexity is likely to increase with the size of the coalition as the complexity of a merge operation can grow significantly with the increase number of candidate nodes in the decoding set of each node. Thus, it takes longer to compute the value/sum-rate of a large coalition compared to a small coalition. However, due to the initial neighbor discovery phase and power costs, most network nodes tend to form coalitions of sizes less than N/2 even for dense networks, under the different power allocation criteria (as will be verified in the following section). VII. SIMULATION RESULTS Consider an ad-hoc network with N =15nodes, where the network density varies with the square area of deployment. The path-loss exponent is set to ν =3, while the correlation coefficient is ρ =.4. The total power constraint per node is P =.15 W, while the noise variance is N =1 5 W. The target SNR for information exchange is set to γ =3dB [1]. The simulation results are averaged over 1 independent runs with the nodes randomly and uniformly distributed across

8 8 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Sum Rate (Bits/s/Hz) Fig. 2. Centralized solution is computationally expensive beyond network density of.5 nodes/unit square area Direct Transmission Distributed Algorithm (EPA) Distributed Algorithm (MMR PA) Distributed Algorithm (SRM PA) Centralized Algorithm (MMR PA with Coalitions) Centralized Algorithm (SRM PA with Coalitions) Centralized Algorithm (MMR PA without Coalitions) Centralized Algorithm (SRM PA without Coalitions) Network Density (Nodes/Unit Square Area) x 1 3 Network Sum-Rate of Different Power Allocation Criteria Average Number of Coalitions Distributed Algorithm (EPA) Distributed Algorithm (MMR PA) Distributed Algorithm (SRM PA) Centralized Algorithm (MMR PA) Centralized Algorithm (SRM PA) Network Density (Nodes/Unit Square Area) Fig. 4. Average Number of Coalitions of Distributed and Centralized Algorithms Average Number of Nodes Per Coalition Distributed Algorithm (EPA) Distributed Algorithm (MMR PA) Distributed Algorithm (SRM PA) Centralized Algorithm (MMR PA) Centralized Algorithm (SRM PA) Network Density (Nodes/Unit Square Area) Fig. 3. Average Number of Nodes Per Coalition of Distributed and Centralized Algorithms the deployment area for different network densities, while the destination is always located at the center of the area. It is evident from Fig. 2 that as the network density increases, the network sum-rate under the different distributed and centralized power allocation criteria also increases and is superior to that of direct transmission 3. This is because with the increase in network density for a fixed number of nodes, the deployment area decreases and the possibility of finding cooperative partners increases. Furthermore, the SRM-PA criterion achieves the highest sum-rate among the other power allocation criteria. Also, the centralized SRM- PA algorithm without coalition formation achieves the highest network sum-rate among all distributed and centralized power allocation criteria 4. Moreover, the computational complexity of the centralized algorithms for network densities beyond.5 nodes/unit square area becomes extremely expensive 5. As can be seen from Fig. 3, the SRM-PA criterion results in the the highest average number of nodes per coalition. This is due to the altruistic coalition formation and the fact that the SRM-PA criterion yields the highest sum-rate. Hence, network nodes tend to form larger coalitions under the SRM- PA criterion to improve the sum-rate of the coalition, which in turn reduces the average number of coalitions formed, as evident from Fig Direct transmission is performed over N time-slots with no cooperation phase. 4 The mathematical formulations and complexity analysis of the centralized power allocation and coalition formation under different power allocation criteria can be found in [21, p. 152]. 5 The centralized MINLP power allocation problems are solved using MIDACO [22] with optimization tolerance set to.1. Average Number of Iterations Fig. 5. Criteria EPA MMR PA SRM PA Network Density (Nodes/Unit Square Area) x 1 3 Average Number of Iterations Under Different Power Allocation In Fig. 5, the average number of iterations until convergence of the proposed distributed merge-and-split algorithm under the different power allocation criteria is shown. It can be seen that the SRM-PA criterion requires the largest number of iterations and this is because under this criterion, network nodes tend to form larger coalitions. Thus, in the proposed distributed merge-and-split algorithm, larger potential coalitions are formed and then possibly split, which in turn increases the number of iterations. Based on the histogram shown in Fig. 6, it can be seen that a large portion of the nodes are participating in coalitions. Even for the EPA criterion, where singletons are most prevalent, more than half of the nodes are participating in coalitions of at least 2 nodes. As for the MMR-PA and SRM-PA criteria, more than half of the nodes are in coalitions of 3 or more nodes, with the SRM-PA criterion resulting in the largest coalitions. In Fig. 7, the mobility of network nodes with density of.5 nodes/unit square area is evaluated, and the network operation is observed over a period of 4 minutes. The mobility of the individual nodes follows the random waypoint (RWP) mobility model [23]. The pause and motion times of each node under the RWP mobility model are uniformly distributed between [, 8] s, and [2, 1] s, respectively. Moreover, the speed is uniformly distributed between s min =.1 m/s and s max, where s min and s max are the minimum and maximum speeds, respectively. As can be seen in Fig. 7, the average number of merge-and-split processes per minute increases with the increase in speed, as expected. This is because the higher is the speed, the more frequent are the network topology

9 BAIDAS and MACKENZIE: ALTRUISTIC COALITION FORMATION IN COOPERATIVE WIRELESS NETWORKS 9 Percentage of Nodes Belongging to Each Coalition Size (%) Coalition Size EPA MMR PA SRM PA Fig. 6. Percentage of Nodes Belonging to Each Coalition Size Under the Proposed Distributed Algorithm - Network Density =.1 Nodes/Unit Square Area Average Number of Merge and Split Processes Per Minute EPA MMR PA SRM PA Maximum Speed s max (m/s) Fig. 7. Average Number of Merge-and-Split Processes Under the Proposed Distributed Algorithm - Network Density =.5 Nodes/Unit Square Area changes, which in turn triggers coalitions to either merge or split more often. Moreover, the SRM-PA criterion requires the highest average number of merge-and-split processes per minute. This is due to the fact that the SRM-PA criterion results in the highest number of coalitions among the different power allocation criteria and thus there is higher tendency to merge or split coalitions in response to network topology changes. Fig. 8 illustrates the average number of coalitions and number of nodes per coalition as a function of time under the different power allocation criteria. As can be seen, the initial network structure starts with 15 singleton coalitions after which network nodes merge (or split) into larger (or smaller) coalitions. More importantly, the average number of coalitions and number of nodes per coalition agree with Fig. 4 (i.e. for the static network). This demonstrates that the proposed merge-and-split algorithm efficiently adapts to the nodes mobility and topology changes. Based on Fig. 5, the proposed merge-and-split algorithm converge for instance under the EPA criterion in about 17 iterations. To allow the proposed algorithm to converge faster and reduce the communication and computational complexities, especially under mobility, the algorithm time-index can be set to a maximum value of. An alternative method to speed up the convergence of the proposed algorithm is to restrict the maximum coalition size to C max. Fig. 9 shows that by reducing the value of without restricting the coalition size (i.e. C max =15), the degradation in the sum-rate is insignificant, even for =5. Similarly, by reducing the maximum coalition size without capping the algorithm time Average Number of Coalitions (EPA) Average Number of Coalitions (MMR PA) Average Number of Coalitions (SRM PA) Average Number of Nodes Per Coalition (EPA) Average Number of Nodes Per Coalition (MMR PA) Average Number of Nodes Per Coalition (SRM PA) Observation Time (s) Fig. 8. Average Number of Coalitions and Number of Nodes Per Coalition Under Proposed Distributed Algorithm - Network Density =.5 Nodes/Unit Square Area and Maximum Speed s max =1.5 m/s Sum Rate (Bits/s/Hz) = 5 = 3 and C max = 5 and C max Network Density (Nodes/Unit Square Area) x 1 3 Fig. 9. Network Sum-Rate Under the Proposed Distributed Algorithm - EPA Criterion with Time-Index and Coalition Size Constraints index (i.e. = ), the sum-rate marginally degrades. In Fig. 1, the average number of iterations for different combinations of time-index and coalition size restrictions is illustrated. Evidently, such constraints significantly reduces the number of iterations at the expense of negligible reduction in the network sum-rate. Finally, Fig. 11 shows the percentage of nodes belonging to each coalition size under the different constraints. It is evident that decreasing the values of and C max prevents large coalitions from forming, which in turn increases the percentage of nodes remaining as singletons and decreases the percentage of nodes forming coalitions of larger sizes. It is noteworthy that based on various network simulations, the performance of the proposed distributed mergeand-split algorithm has been found to usually lag that of the centralized algorithm, suggesting that there is no D c - stable partition in most cases (i.e. in these cases, the mergeand-split algorithm finds a suboptimal D hp -stable partition instead). In other words, in any network instance in which the centralized algorithm returns a better solution than the distributed merge-and-split algorithm, one can say for sure that there is not a D c -stable partition (because if there were a D c -stable partition, then it would have been optimal and the merge-and-split algorithm would have found it, so it would be impossible for the centralized algorithm to return a better solution). Therefore, our numerical results suggest that D c -stable partitions rarely exist in the simulated network instances.

10 1 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Average Number of Iterations = 5 = 3 and C max = 5 and C max Network Density (Nodes/Unit Square Area) x 1 3 Fig. 1. Average Number of Iterations Under the Proposed Distributed Algorithm - EPA Criterion with Time-Index and Coalition Size Constraints Percentage of Nodes Belonging to Each Coalition Size (%) = 5 = 3 and C max = 5 and C max Coalition Size Fig. 11. Percentage of Nodes Belonging to Each Coalition Size Under the Proposed Distributed Algorithm - EPA Criterion with Time-Index and Coalition Size Constraints - Network Density =.1 Nodes/Unit Square Area In [21, p. 157], an example ad-hoc network with N =1 nodes and specific network topology is presented. Specifically, the initial and final network partitions under the centralized and distributed algorithms with different power allocation criteria are studied. Additionally, the resulting network sumrates are summarized. VIII. DISCUSSION A. Network Coding As it is the case with common communication systems, our system model is based on TDMA, where each of the N network nodes is assigned a time-slot for transmission of its data. Additionally, there is a single time-slot that is allocated for multiple-access cooperative transmission. Therefore, the total number of time-slots is N +1. The multiple-access cooperative transmission has been made possible via the use of network-coded transmission and multiuser detection at the destination node. In the case where the merge-and-split algorithm converges with no formed coalitions of two or more nodes, then the multiple-access time-slot will not be utilized and is thus wasted. In such case, the achievable network sumrate is less than that of direct transmissions sum-rate. This happens when the network nodes are far away from each other (i.e. network density is low). On the other hand, when the network nodes are within close proximity of each (i.e. network density is high), then coalitions of two or more nodes are more likely to be formed. In this case, the multiple-access time-slot is utilized and the achievable network sum-rate is greater than the sum of the direct transmission rates. In conventional TDMA-based relay communications (without network-coding and multiuser detection), the total number of time-slots for a network of K coalitions (where 1 K N) is determined as [1] K T = N + I(C k ) C k ( C k 1), (23) k=1 where I(C k )=1if C k 2, and otherwise. In the case of singletons, T = N. However, in the case of a grand coalition, then T = N 2. For instance, when K = N 1 (i.e. there is a coalition of size 2), then T = N+2. Clearly, our system model is thus more bandwidth efficient than conventional cooperative communication systems. B. Coalition-Head Selection A coalition-head can be selected randomly by the nodes in the coalition (by exchanging an advertisement message). For instance, in the coalition formation phase, all the nodes in the coalition could select a coalition-head by selecting a random number between and 1. The node with the smallest/largest number is then selected as coalition-head [24]. Coalition-head selection may also be based on a node s attributes such as their identification number, location or available transmission or processing resources [25] [26]. It should be noted that the coalition-head is responsible for periodically broadcasting timing information to the rest of the coalition and determining the power allocation fractions (which are convex optimization problems see Section V). Such tasks do not pose severe computational overhead on the coalition-head. However, for fairness, the task of being a coalition-head can be performed in a round-robin fashion. Interested readers can also refer to [8,9], where message exchanges for coalition-head selection and coalition formation are discussed. C. Timing Synchronization In practice, for distributed timing synchronization, the coalition-head is responsible for exchanging timing information (i.e. the SYNC signal) through periodic beacon transmissions via appropriate control channels. The other nodes synchronize their clocks according to a time-stamp in the beacon [15] [16]. It should be noted that mismatches in clocks of the geographically distributed nodes results in mis-synchronized transmissions. However, due to the target SNR requirement within each coalition, only a small number of nodes within close proximity of each other form a coalition. Therefore, it is reasonable to assume that the nodes within a single coalition are perfectly synchronized with the coalition-head. In this case and from a network perspective, each coalition can be considered as a single entity and thus there is missynchronization between the coalitions and the destination node. However, this case is beyond the scope of this work as our focus has been to study distributed altruistic coalition formation and the impact of different power allocation criteria. Intuitively, it is expected that the achievable rate of each node under imperfect timing synchronization to degrade.

11 BAIDAS and MACKENZIE: ALTRUISTIC COALITION FORMATION IN COOPERATIVE WIRELESS NETWORKS 11 D. Mobility In mobile ad-hoc wireless networks (MANETs), nodes are mobile and thus topology changes are inevitable. In such case, nodes move in and out of the communication range of other nodes, leading coalitions to be physically separated and communication links to fail. In [27], a cooperation model has been proposed based on coalitional game theory to incentivize cooperation, while trying to restore average node reachability upon topology changes caused by mobility. Particularly, the proposed model maintains and restructures the formed coalitions to restore reachability and stability. The future work in this direction aims at studying coalition formations for many-to-many cooperative communications while incorporating reachability under different individual and group mobility models. E. Power Allocation Although this paper has analyzed and compared the impact of different power allocation criteria on coalition formation, the selection of the power allocation criterion is applicationspecific. For instance, if the aim is to maximize the network sum-rate, then the SRM-PA criterion will be utilized throughout the network operation. Moreover, if the aim is to maximize the minimum rate of each node in the network so as to achieve rate-fairness, then the MMR-PA criterion is used. Finally, if the aim is to allocate power with minimal communication overhead and computational complexity and no specific rate requirement, then the EPA criterion is used. IX. CONCLUSIONS In this paper, altruistic coalition formation for cooperative relay networks has been studied. A distributed merge-and-split algorithm has been designed based on the utilitarian order and under different cooperative power allocation criteria. It has been shown that the proposed algorithm allows network nodes to self-organize into disjoint coalitions and that the sum-of-rates maximizing power allocation criterion results in the largest average coalition size and number of nodes per coalition, among the different power allocation criteria. Centralized power allocation and coalition formation are also investigated, where it shown that the proposed algorithm achieves a network sum-rate that is comparable with that of a centralized controller; however, with less computational complexity. Finally, the proposed algorithm has been shown to efficiently adapt to nodes mobility and network topology changes. REFERENCES [1] K. J. R. Liu, A. K. Sadek, W. Su, and A. Kwasinski, Cooperative Communications and Networking. Cambridge University Press, 28. [2] W. Saad, Z. Han, M. Debbah, and A. 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12 12 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Mohammed W. Baidas (S 5-M 12) received the B.Eng. (first class honors) degree in communication systems engineering from the University of Manchester, UK and the M.Sc. degree with distinction in wireless communications engineering from the University of Leeds, UK, and also the M.S. degree in electrical engineering from the University of Maryland, College Park, USA in 25, 26 and 29, respectively. He earned his Ph.D. in electrical engineering at Virginia Tech, USA in 212. Dr. Baidas is currently an Assistant Professor in the Electrical Engineering Department at Kuwait University. His research focuses on resource allocation and management in cognitive radio systems, game theory, and cooperative communications and networking. Allen B. MacKenzie (SM 8) received his bachelor s degree in Electrical Engineering and Mathematics from Vanderbilt University in In 23, he earned his Ph.D. in electrical engineering at Cornell University and joined the faculty of the Bradley Department of Electrical and Computer Engineering at Virginia Tech, where he is now an associate professor. During the academic year, he was an E. T. S. Walton Visiting Professor at Trinity College Dublin. Prof. MacKenzie s research focuses on wireless communications systems and networks. His current research interests include cognitive radio and cognitive network algorithms, architectures, and protocols and the analysis of such systems and networks using game theory. His past and current research sponsors include the National Science Foundation, Science Foundation Ireland, the Defense Advanced Research Projects Agency, and the National Institute of Justice. Prof. MacKenzie is a senior member of the IEEE and a member of the ASEE and the ACM. Prof. MacKenzie is an associate editor of the IEEE Transactions on Communications and the IEEE Transactions on Mobile Computing. He is the author of more than 45 refereed conference and journal papers and the co-author of the book Game Theory for Wireless Engineers.